Related papers: On sharp bounds of certain Close-to-Convex functio…

Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction…

We consider the family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. Our objective in this paper is to estimate the difference of the moduli of successive coefficients, that is $\big |…

Recently, Charpentier showed that there exist holomorphic functions $f$ in the unit disk such that, for any proper compact subset $K$ of the unit circle, any continuous function $\phi$ on $K$ and any compact subset $L$ of the unit disk,…

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0) = 0$, and $f'(0) = 1$. Let $F$ be the inverse function of $f$, given by $F(w)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some…

This paper deals with an analog of the Mahler volume product related to the ${\cal J}$ transform acting in the class of geometric convex functions ${\rm{Cvx}}_0({\mathbb R}^n)$. We provide asymptotically sharp bounds for the quantity…

Inspired by the recent works of Srivastava et al. (HMS-AKM-PG), Frasin and Aouf (BAF-MKA) and others (Ali-Ravi-Ma-Mina-class,Caglar-Orhan,Goyal-Goswami,Xu-HMS-AML,Xu-HMS-AMC), we propose to investigate the coefficient estimates for a…

Let $\ID$ denote the open unit disc and let $p\in (0,1)$. We consider the family $Co(p)$ of functions $f:\ID\to \overline{\IC}$ that satisfy the following conditions: \bee \item[(i)] $f$ is meromorphic in $\ID$ and has a simple pole at the…

We give some coefficient bounds and distortion theorems for a subclass of univalent functions in the unit disk, and defined using the S\^{a}l\^{a}gean differential operator. The results generalize and unify some well known results for…

We show that the Carath\'{e}odory number of the joint numerical range of $d$ many bounded self-adjoint operators is at most $d-1$, and even at most $d-2$ if the underlying Hilbert space has dimension at least $3$. This extension of the…

Let $\mathcal{A}$ be the class of all analytic functions $f$ defined on the open unit disk $\mathbb{D}$ with the normalization $f(0)=0=f^{\prime}(0)-1$. This paper examines the radius of concavity for various subclasses of $\mathcal{A}$,…

We show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\R^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex…

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

We give a class of domains for which Fridman invariant and injectivity radius function coincide with respect to Carath\'eodory metric. We give explicit expressions of the squeezing functions for these domains and investigate some of their…

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of…

In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of…

In this paper, we investigate three specific subclasses of Ma-Minda type convex functions: namely, convex functions of order $\alpha$, Janowski convex functions, and Robertson functions of normalized analytic functions defined in the open…

Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ We examine the properties of the class $\mathcal{C}(\varphi)$ defined as…

The approximate Carath\'eodory theorem states that given a compact convex set $\mathcal{C}\subset\mathbb{R}^n$ and $p\in\left[2,+\infty\right[$, each point $x^*\in\mathcal{C}$ can be approximated to $\epsilon$-accuracy in the $\ell_p$-norm…

Sharp upper and lower bounds for the second and third order Hermitian-Toepilitz determinants are obtained for some generalized subclasses of starlike and convex functions. Applications of these results are also discussed for several widely…

Let ${\mathcal S}$ denote the class of all functions $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$ analytic and univalent in the unit disk $\ID$. For $f\in {\mathcal S}$, Zalcman conjectured that $|a_n^2-a_{2n-1}|\leq (n-1)^2$ for $n\geq 3$. This…